Analytic properties of the sum of a power series

Let the power series $\sum_{n=0}^{\infty } c_{n} z^{n}$, with positive radius of convergence $R$, represent the function $f$ in the disk $\left|z\right|<R$. For $k=0,1,\ldots$ let $s_{k}$ be the $k$th partial sum of the series $s_{k}(z) = \sum_{n=0}^{k} c_{n} z^{n}$. Prove that

for each $z$ in the disk $\left|z\right|<R$.

Let the function $f$ be analytic in the open unit disc of the complex plane and real valued on the radii $[0,1]$ and $[0, e^{i\pi \sqrt{2}}]$. Prove that $f$ is constant.

Prove that the power series $\displaystyle \sum_{n=0}^{\infty } a_{n}x^{n}$ converges uniformly on any closed interval lying entirely within its interval of convergence.

Does there exist an analytic function in $\left|z\right| < 1$ such that $0 < \left|f(\frac{1}{n})\right| < e^{-n}$ for $n = 2, 3, 4, \ldots$? Justify your answer.

Let $D = \left\{ z \in \mathbb {C} : \left|z\right| < 1\right\}$. Find all holomorphic functions $f : D \to \mathbb {C}$ such that $f\! \left(\frac{1}{n} + ie^{-n}\right)$ is real for all integers $n \geq 2$.

Find by direct differentiation the derivative of the series $x-\dfrac {x^{2}}{2}+\dfrac {x^{3}}{3}-\dfrac {x^{4}}{4}+\ldots$, $\left|x\right| < 1$.

Find by direct differentiation the derivative of the series $x-\dfrac {x^{3}}{3}+\dfrac {x^{5}}{5}-\dfrac {x^{7}}{7}+\ldots$, $\left|x\right| < 1$.

Find by direct differentiation the derivative of the series $x-\dfrac {x^{3}}{3!}+\dfrac {x^{5}}{5!}-\dfrac {x^{7}}{7!}+\ldots$.

Find by direct differentiation the derivative of the series $1+x+\dfrac {x^{2}}{1 \cdot 2}+\dfrac {x^{3}}{1 \cdot 2 \cdot 3}+\dfrac {x^{4}}{1 \cdot 2 \cdot 3 \cdot 4}+\ldots$.

Let $\left(c_{k}\right)_{k \geq 0}$ be a fixed sequence of real numbers. Show that if the series $\sum_{k = 0}^{\infty } c_{k} y^{k}$ converges for some $y \in \mathbb {R}$, then the series $f(x) = \sum_{k = 0}^{\infty } c_{k} x^{k}$ converges absolutely for all $\left|x\right| < \left|y\right|$, and show that this function $f$ is infinitely differentiable for all $x$ with $\left|x\right| < \left|y\right|$.

Prove that the power series $\sum_{n = 0}^{\infty } a_{n}\left(x-x_{0}\right)^{n}$ converges uniformly on any segment lying inside its interval of convergence.

Prove that if the function $f(z)$ can be represented in the disk $\mid z$ - $z_{0} \mid < R$, where $R > 0$, by a power series $f(z) = \sum_{n = 0}^{\infty } c_{n}\left(z-z_{0}\right)^{n}$ converging to it, then:

Let the power series

be convergent for $x=1$ and assume $0<\alpha <1$. Then the power series

is convergent for $h=1-\alpha$.

Let

be arbitrary functions, $\left|a_{nk}\right|<A$ for all positive integral values of $n$ and $k$ and

Then we have for fixed $k$, $k=1,2,3,\ldots$

Let $\lim_{n \rightarrow \infty } \sqrt[n]{\left|a_{n}\right|} = 1$. Prove the uniform convergence of the series on the set $E$:

Prove that the power series $\sum_{n = 1}^{\infty } c_{n} z^{n}, \quad \sum_{n = 1}^{\infty } \frac{c_{n}}{n+1} z^{n+1}$ and $\sum_{n = 1}^{\infty } n c_{n} z^{n-1}$ have the same radius of convergence.

Give a proof of Proposition A.2.6 (vi) (term by term differentiability of a power series) using Proposition A.2.7 (iv) (the fundamental theorem of Riemann integration).